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CEMC Cayley

Algebra Difficulty 2.1 Junior Find the answer

The operation a∇ba \nabla b is defined by a∇b=a+ba−ba \nabla b=\frac{a+b}{a-b} for all integers aa and bb with a≠ba \neq b. If 3∇b=−43 \nabla b=-4, what is the value of bb?

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Using the definition, 3∇b=3+b3−b3 \nabla b=\frac{3+b}{3-b}. Assuming b≠3b \neq 3, the following equations are equivalent: 3∇b=−43 \nabla b =-4, 3+b3−b=−4\frac{3+b}{3-b} =-4, 3+b=−4(3−b)3+b =-4(3-b), 3+b=−12+4b3+b =-12+4 b, 15=3b15 =3 b and so b=5b=5.

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